Set-valued games and mixed-strategy equilibria in discounted supergames

Abstract

This paper examines the subgame-perfect mixed-strategy equilibria in discounted supergames. We present a method that finds all the equilibrium payoffs without public correlation in 2 × 2 games. The method makes it possible to solve and analyse the set of equilibria in more detail. This is the first time that the mixed-strategy payoff set is solved in the repeated prisoner’s dilemma. We find that the players can obtain higher payoffs in mixed strategies, there are more Pareto efficient outcomes, and the set of equilibria can be dramatically larger. We show that the equilibrium payoffs can be efficiently computed by finding certain extreme points of the set. This result relies on the classification of games, the monotonicity properties of the problem and splitting the sets into X-Y convex parts. We also introduce set-valued games, where the players’ payoffs are chosen from given sets, and show that these games can be solved with the same methodology.